Biomedical terminologies and ontologies are enabling resources for clinical decision support systems and data integration systems for translational research and health analytics. Therefore, the quality of these resources has a direct impact on healthcare and biomedical research. In the past decade, quality assurance (QA) of biomedical terminologies has become a key issue in the development of standard terminologies and has emerged as an active field of research. Approaches to quality assurance include the use of lexical, structural and semantic techniques applied to biomedical terminologies, as well as techniques for comparing and contrasting these resources. As part of the Medical Ontology Research project, explore quality assurance and interoperability issues in a variety of biomedical terminologies including drug terminologies, clinical terminologies, and specialized terminologies, such as HPO the Human Phenotype Ontology and the Orphanet terminology for rare diseases. About half of our investigations have a primary focus on quality assurance, for which we developed novel methods. In the other half, we apply existing techniques to assess interoperability among terminologies or some aspect of quality (e.g., coverage) in a terminology. In our work, we put special emphasis on the development of principled, automated, scalable methods, applied systematically to the entire content of a terminology by independent researchers, as opposed to manual review of subsets by domain experts. The QA processes we develop have proved effective in identifying a limited number of errors that had defeated the quality assurance mechanisms in place in terminology development systems. We share our findings and techniques with the scientific community through scientific publications and presentations at conferences. Whenever possible, we also report these issues to the developers of the biomedical terminologies we investigated. This project is also a contribution to the LHC Training Program, because many investigations involve post-doctoral fellows or summer (graduate and undergraduate) students.